A woman cycling alone in the Middle East will undoubtedly be the centre of attention but should be quite safe with a few basic precautions.
In general the rule of hospitality towards visitors applies to a single woman as much as it does to a man or a couple. However, some attitudes towards women still persist in the Middle East that Westerners will find hard to handle.
Some of these behaviours and perceptions are harmless, if a little overwhelming. People will find it quite exceptional to see a woman on a bicycle at all, let alone travelling unaccompanied, so you can expect double the amount of attention and invitations to share a cup of tea that other cycle tourists would get. Local police may an extra effort to take care of you by escorting you through their area.
On the downside, a minority of men with misguided ideas about Western women (mostly formed by Hollywood movies) may try things they would not dare to do with women from their own culture. This can go from mild flirting to groping.
Be aware that maintaining eye contact or smiling at someone of the opposite sex is seen as flirting and can be wrongly taken as a signal to go further. You won’t have this problem with men who are with their families but you should be wary of single men such as truck drivers you encounter on the road. If you have the choice, ask a woman for help or directions.